Play already stops for an injury, so why not review it at the time?
Because I think reviews would result in the worst of both worlds -- they won't stop the fakes and they'll add yet more and longer interruptions of the game. Which is what the fakers want, which will lead to more fakes than we already have.
Here's why:
We tend to remember the blatantly obvious fakes -- players walking around naturally, coaches yelling and gesticulating from the sidelines to go down, and the player suddenly hits the turf like he's been shot with a load of 00-buck. Those are the easy ones to identify, and replay would definitely catch them.
Coaches, however, are remarkably adaptive when it comes to getting around rules. If you introduce consequences for faking (minimum sit-outs, yardage penalties, docked time-outs, mandatory clock runoff, whatever), you've only given them an incentive to improve the acting.
In pretty short order, the fakes will be infinitely less obvious than the poster-child film clips we've all seen. For anybody who thinks coaches wouldn't do that, I have an old bridge in Brooklyn that I need to sell for cheap...you know, for tax purposes.
So if you institute film review, you've now put the refs in the impossible position of getting inside the head of the player. Since the fakes will now be far less obvious, most if not all of the reviews will be inconclusive.
IOW, they won't have clear evidence that it's a fake, so they won't be able to assess whatever consequences have ostensibly been attached to fake injuries.
Delay is what the fakers want -- time to gather themselves, call the play (offensive or defensive), make substitutions, etc. all while the game clock is stopped -- and replay gives them more of all of that. Worse, you now have delays for both the injury (fake or genuine) and the review.
Which, because the acting is so much better, thereby preventing assessment of consequences, is more incentive to fake more often.
We all want to end the fake injuries. Film review puts an impossible burden on the refs and before the first kickoff will actually be counter-productive.
A simple rule -- if you're hurt bad enough that we have to stop the game to tend to your injury, you're hurt bad enough to sit out a minimum of the rest of the possession -- is easy to implement, easy to enforce and will be effective.